The Leslie–Gower model with weak Allee effect is given by (REF ) with \(m<0\) , and we only consider the model in the domain \(\Omega =\lbrace \left(N,P\right)\in \mathbb {R}^2,N>0, P\ge 0\rbrace \)  and \(\left(r,K,q,a,s,h\right)\in \mathbb {R}{}{}_{+}^{6}\) , \(m<0\)  and \(a<K\) . As system (REF ) is of Kolmogorov type [1]} since \(dN/dt=N\cdot W\left(N,P\right)\)  and \(dP/dt=P\cdot R\left(N,P\right)\) . That is, the axes are invariant. The positive equilibrium points of the systemIn this manuscript, we will refer to an equilibrium point as with two positive entries as a positive equilibrium point. (REF ) are \(\left(K,0\right)\)  and \(\left(x^*,y^*\right)\)  which is the intersection of the nullclines
\(P=hN \quad \text{and} \quad P=\dfrac{r}{q}\left( 1-\dfrac{N}{K}\right)\left(N+a\right)\left(N-m\right).\) 
